Multiscale stabilization for convection diffusion equations with heterogeneous velocity and diffusion coefficients

TL;DR

This paper introduces a new multiscale stabilization method for convection-diffusion equations with heterogeneous coefficients, improving stability and accuracy especially at high Peclet numbers.

Contribution

The paper proposes a novel stabilization technique combining constraint energy minimization and discontinuous Petrov-Galerkin formulation for multiscale problems.

Findings

Test functions exhibit localization, enabling local computation.

Method achieves stability with solution errors close to best approximation.

Numerical results validate the effectiveness of the stabilization.

Abstract

We present a new stabilization technique for multiscale convection diffusion problems. Stabilization for these problems has been a challenging task, especially for the case with high Peclet numbers. Our method is based on a constraint energy minimization idea and the discontinuous Petrov-Galerkin formulation. In particular, the test functions are constructed by minimizing an appropriate energy subject to certain orthogonality conditions, and are related to the trial space. The resulting test functions have a localization property, and can therefore be computed locally. We will prove the stability, and present several numerical results. Our numerical results confirm that our test space gives a good stability, in the sense that the solution error is close to the best approximation error.